PHYSICAL REVIEW D 102, 075019 (2020)
Simple hidden sector dark matter
Patrick Barnes ,1 Zachary Johnson ,1 Aaron Pierce ,1 and Bibhushan Shakya 2 1Leinweber Center for Theoretical Physics, Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA 2CERN, Theoretical Physics Department, 1211 Geneva 23, Switzerland
(Received 13 June 2020; accepted 16 August 2020; published 16 October 2020)
A hidden sector that kinetically mixes with the minimal supersymmetric standard model provides simple and well-motivated dark matter candidates that possess many of the properties of a traditional weakly interacting massive particle (WIMP). These supersymmetric constructions can also provide a natural explanation for why the dark matter is at the weak scale even if it resides in a hidden sector. In the hidden sector, a natural pattern of symmetry breaking generally makes particles and their superpartners lie around the same mass scale, opening novel possibilities for a variety of cosmological histories and complex indirect detection signatures.
DOI: 10.1103/PhysRevD.102.075019
I. MOTIVATION SM and hold no direct connection to a hierarchy problem solution. In particular, the WIMP miracle can be realized A thermally produced stable particle with a weak scale with order one couplings within a separate weak scale dark annihilation cross section reproduces the observed dark sector that only interacts very feebly with the SM. This idea matter relic abundance [1]. The weak scale is known to be was dubbed secluded dark matter in [9]. The existence of an important scale from the particle physics perspective, such secluded/hidden/dark sectors, with extended gauge and many beyond the Standard Model (BSM) theories groups, is well motivated from a string theory perspective, have dark matter (DM) candidates at this scale, often as and their interactions with the SM can give rise to several part of a solution to the hierarchy problem. This coinci- interesting phenomenological signatures (see [10] and dence has been dubbed the weakly interacting massive references therein). In particular, the kinetic mixing portal particle (WIMP) miracle, and thermal DM associated with [11], where a gauged Uð1Þ0 in a hidden sector mixes with hierarchy problem solutions has been the subject of intense the SM hypercharge Uð1Þ [12], has been extensively theoretical research as well as experimental searches. An Y studied in the literature, including in the context of dark example is the lightest supersymmetric particle (LSP) [2], matter that realizes its thermal abundance via the afore- stabilized by R-parity. However, even though several mentioned “WIMP” miracle, see, e.g., [9,13–16]. compelling arguments—ranging from gauge coupling uni- If the DM is near the weak scale but in a separate sector, fication to considerations of the underlying theory of it is of interest to understand how that sector knows about quantum gravity—provide reasons to believe that super- the weak scale. In supersymmetric theories, this may occur symmetry is part of the underlying description of nature, naturally if supersymmetry breaking is mediated to both the absence of signals at DM direct detection experiments sectors with approximately equal strength, as might hap- such as Xenon1T [3] and the nonobservation of super- pen, e.g., in theories of gravity mediation. In this case, the partners at the Large Hadron Collider (LHC) have led to masses in the two sectors are correlated at some UV scale questions about whether the simplest instantiation of this but may be separated at lower energies by running effects. WIMP DM idea is realized in nature [4–6]. We use this line of argument to motivate hidden sector While it is possible that a weak scale cross section is spectra. While hidden sector particles such as heavy Z0’s associated with the Standard Model (SM) weak interactions are difficult to probe directly,1 the existence of such a themselves [7,8], it is not necessary. New dynamics (largely) hidden sector could have consequences for cos- associated with the DM particle may be unrelated to the mology, in particular for dark matter. Here we will work under the assumption that the two sectors are coupled strongly enough that the hidden and visible sectors Published by the American Physical Society under the terms of thermalize. the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, 1For studies of possible phenomenological implications of and DOI. Funded by SCOAP3. hidden sector gauge bosons and their superpartners, see [17,18]).
2470-0010=2020=102(7)=075019(19) 075019-1 Published by the American Physical Society BARNES, JOHNSON, PIERCE, and SHAKYA PHYS. REV. D 102, 075019 (2020)
While this broad-brush picture has some appeal, it is of emerge from reasonable parameter choices in the UV. interest to ask whether the data from the LHC can tell us Section VII is devoted to the discussion of the decay more about such a supersymmetric setup. The absence of modes of various hidden sector particles. Direct detection superpartners at the LHC suggests the weak scale itself is and collider constraints are explored in Sec. VIII, followed somewhat fine-tuned, as does the Higgs boson mass, which by a discussion of indirect detection signals in Sec. IX.We requires large loop level corrections due to supersymmetry end with some concluding remarks in Sec. X. breaking [19–21]. In fact, the relatively large value of the observed Higgs boson mass suggests that the scale of II. A SIMPLE DARK SECTOR supersymmetry breaking is several TeV in the absence of In addition to the field content of the MSSM, we significant stop mixing in the minimal supersymmetric consider a dark/hidden sector with gauge group Uð1Þ0, “ ”— Standard Model (MSSM). This little hierarchy problem gauge coupling g0, and a trio of SM-singlet superfields: a wherein the weak scale is tuned at a subpercent level—might dark Higgs field Hˆ 0 with charge Q0 ¼þ1 (which breaks the simply be accidental, or find explanations in some anthropic Uð1Þ0 symmetry once the scalar component obtains a vev), or cosmological selection process. None of these need apply a superfield Tˆ with charge Q0 ¼ −1 (necessary for can- to the hidden sector, and as a consequence the vacuum cellation of anomalies related to Uð1Þ0), and a singlet expectation value (vev) in the hidden sector should be more ˆ 0 0 closely tied to the scale of supersymmetry breaking. superfield S with charge Q ¼ (necessary to enable a ˆ 0 ˆ 0 WIMP dark matter in minimal supersymmetric setups is scale-invariant superpotential involving H , T ). The most 1 0 made stable by assuming R-parity. This, however, does not general superpotential after imposing the above Uð Þ ˆ work for a hidden sector dark matter candidate if the hidden charges along with any symmetry under which both S sector spectrum is heavier than that of the visible sector, and Tˆ transform nontrivially is since the said dark matter candidate can decay into the W λˆ ˆ ˆ 0 visible sector even in the presence of R-parity. In this case, hid ¼ S T H : ð1Þ DM can instead be stabilized by other, perhaps accidental, symmetries realized in the hidden sector. Interestingly, This superpotential possesses a Z2 symmetry under which R-parity need not be conserved, and the breaking of both Sˆ and Tˆ are odd; this ensures the lightest particle in the ˆ ˆ R-parity might even be desirable, for instance, to break S − T system, which we will refer to as the lightest Z2 odd baryon number in order to realize baryogenesis, as particle (LZP), is stable and therefore a dark matter studied in [22]. candidate.3 We assume that the hidden sector communi- In this paper, we combine the above ideas to construct cates with the visible sector via supersymmetric kinetic simple and realistic models for hidden sector dark matter. mixing [29]: We assume this sector interacts with our own via super- Z symmetric kinetic mixing. All the ingredients—hidden ϵ d2θW W0 : : sectors, supersymmetry, and kinetic mixing—are well 2 Y þ H c motivated. In the absence of accidental tuning in the hidden ϵ ϵ 0 − μν 0 ϵ ˜ σμ∂ ˜ 0† ϵ ˜ 0σμ∂ ˜ † sector, no large mass hierarchies are expected between ¼ DYD 2FY Fμν þ i B μB þ i B μB ; ð2Þ hidden sector particles and their superpartners, so that a 0 multitude of particles can be involved in both dark matter where the WY, W represent the chiral field strength freeze-out as well as present day dark matter annihilation.2 multiplet for Uð1Þ hypercharge and the hidden sector Y ˜ Our study therefore illuminates the wide range of dynamics Uð1Þ0, respectively, and we use the notation B0 for the that can give rise to a WIMP-like miracle in well-motivated hidden sector gaugino. This basic set-up has previously hidden sectors. also been considered in the context of asymmetric dark The outline of the paper is as follows: In Sec. II, matter [30], as well as a way to generate dark matter at the we describe the field content of the hidden sector we GeV scale [31]. It was also considered in some detail in consider, outlining the possible dark matter candidates. [32], where some consequences for thermal histories and This is followed by detailed studies of fermion and scalar direct detection were considered. dark matter in Secs. III and IV, respectively. Section V We assume supersymmetry breaking induces a vev for pffiffiffi addresses cases where the mass gap between the dark H0 only, hH0i¼v0= 2. This vacuum will be preferred matter candidate and its superpartner is small, leading to when there is either a hierarchy between the soft masses coannihilation effects and long lifetimes. We then explore for the scalars, or if λ is large enough to overcome the relations between parameters in the UV and IR in Sec. VI, D-flatness condition (which favors hH0i¼hTi). This vev discussing how consistent cosmological histories can 3In contrast, a pure singlet superfield Sˆ that can couple to SM 2For related studies of dark matter in very supersymmetric fields would give rise to decaying dark matter via a neutrino hidden sectors, see [23]. portal, see e.g., [24–28].
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0 0 0 provides the hidden gauge boson Z with a mass mZ0 ¼ g v between superpartners (i.e., because we assume less tuning) and combines the fermion components of Sˆ and Tˆ into a in the hidden sector, and since large values of λ in the UV pffiffiffi pffiffiffi 0 λ 2 0 Dirac fermion, which we denote ψ, with mψ ¼ λv = 2. will rapidly flow to the fixed point ¼ g , causing the For convenience, we define as usual α0 ≡ g02=4π and Yukawa and gauge corrections to the hidden sector Higgs 2 0 mass to partially cancel. The corrections can become αλ ≡ λ =4π. As we will discuss later (Sec. VI), αλ ¼ 2α λ is an RG fixed point where an accidental N ¼ 2 SUSY is significant for large values of , particularly in the case λ ≫ 0 restored; at this point the ψ, Z0, and H0 are degenerate. g , but we note that this occurs in a region of parameter The hidden neutralino sector has the following mass space where fine-tuning in the hidden sector is severe. In ˜ ˜ matrix in the B0, H0 basis: what follows, we therefore generally assume the tree level relation m 0 ¼ m 0 , and comment on places where this H Z assumption may fail. Finally, the kinetic mixing induces m ˜0 m 0 B Z 0 0 Mχ0 ¼ ; ð3Þ 0 corrections to the mass eigenvalues of both Z and H ,but mZ0 these are generally quite small. In the hidden scalar sector, the supersymmetry breaking with m ˜0 the hidden sector soft supersymmetry breaking B soft terms are gaugino mass parameter. The off-diagonal terms arise when 0 ˜ → 0 the H takes on its vev. In the limit mB0 (also taking L ⊃ ˜ 2 ˜ 2 ˜ 2 ˜ 2 ˜ 2 ˜ 0 2 λ ˜ ˜ ˜ 0 ˜ ˜ mSjSj þ mTjTj þ mH0 jH j þð AλS T H þ H:c:Þ: ϵ → 0), B0 pairs with H0 to form a Dirac neutralino that is 0 ˜ ð6Þ degenerate with Z . A nonzero mB0 splits this state into two 0 0 Majorana mass eigenstates, which we denote χ1 and χ2, In the (S;˜ T˜ ) basis, the scalar mass matrix can be written as 0 0 ˜ 0 with mχ1 075019-3 BARNES, JOHNSON, PIERCE, and SHAKYA PHYS. REV. D 102, 075019 (2020) 0 (ii) With RPC, an interesting wrinkle occurs if χ1 is type may be constructed, significant dilution would 0 sufficiently long-lived (due to small ϵ). In this case, χ1 spoil the “WIMP miracle” that this scenario realizes. can freeze-out prior to the LSP, but decay after LSP In the remainder of this paper, we only consider freeze-out, contributing another secondary LSP DM scenarios where R-parity is broken. Thus both the LZP 0 abundance (since each χ1 decay produces an LSP). In superpartner and the LSP are unstable, and the LZP is the 0 principle, the DM from χ1 decay might give a too- sole DM candidate. In the next two sections, we discuss a large DM abundance. However, such concerns are variety of possible spectra with fermion and scalar LZP 0 ˜ mitigated because the χ1 − B coannihilation process, dark matter, respectively, addressing cosmological histories 0 which can determine χ1 freeze-out, while ϵ sup- and present day annihilation cross sections. pressed, can remain in equilibrium longer than naively III. FERMION DARK MATTER expected because of the relatively unsuppressed B˜ abundance. The result is that it is not difficult to We first review the cosmology of fermion LZP ψ DM χ0 suppress the 1 freeze-out abundance (and hence the freeze-out with simplified analytic expressions to under- secondary LSP abundance) to acceptable levels. stand the broad picture, followed by detailed numerical (iii) Another interesting possibility in the RPC case is the treatment to include more complicated cases. existence of a trio of dark matter states. If the mass ψ LZPs can annihilate via s-wave processes within the splitting between the LZP and its superpartner is dark sector unless ψ is the lightest dark sector state. Over smaller than the LSP mass, then decays between the much of the parameter space, the Z0H0 channel dominates if 0 0 two are kinematically forbidden. The LZP, its super- open (αλ > 2α ); recall that αλ ¼ 2α is an RG fixed point. partner, and the LSP are thus all stable components The H0H0 channel is p-wave suppressed, and the Z0Z0 0 0 0 5 of dark matter. channel is suppressed by α =αλ relative to Z H . When (iv) On the other hand, if R-parity is broken, the LSP Z0H0 is kinematically forbidden, the only other channel decays into SM particles via RPV interactions. For potentially available completely within the dark sector is 0 0 0 0 0 concreteness, consider the baryon number violating χ1χ1. Annihilations to Z H proceed either via s-channel Z coupling: exchange or t=u-channel ψ exchange, while annihilations 0 0 0 to χ1χ1 proceed either via s-channel Z or t=u-channel λ″ c c c WRPV ¼ ijkUi Dj Dk: ð8Þ scalar exchange. In the limit where the scalars are decoupled, the annihilation cross sections are λ″ If is small, the consequently long lifetime of the 2 qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 πα0 16ηχ0 þ 2η 0 −ηχ η 0 ð8 þ η 0 Þ LSP is a potential concern, since the LSP abundance σ ≈ 4 1 − η 1 Z 1 Z Z h viχ0 χ0 cθ 2 χ0 2 2 ; 1 1 N 1 η η −4 can grow to dominate the energy density of the mψ Z0 ð Z0 Þ Universe, and the significant entropy from its ð9Þ subsequent decays may dilute the abundance of LZP dark matter. While viable cosmologies of this 2 1=2 2 3 4 5 πα ð1 − η 0 Þ ð64 − 128η 0 þ 104η 0 − 30η 0 þ η 0 þ η 0 Þ σ ≈ λ Z Z Z Z Z Z h viZ0H0 2 2 2 ; ð10Þ 4mψ ð2 − ηZ0 Þ ð4 − ηZ0 Þ 2 2 0 where η 0 ≡ m 0 =mψ ¼ 2α =αλ, so that η 0 ¼ 1 represents 2 2 Z Z Z mψ 0 07 η ≡ 2 2 θ Ω 2 ≈ Ω 2 . the IR fixed point; χ0 mχ0 =mψ , and N is the neutralino ð h ÞZ0H0 DMh ; ð12Þ 1 1 1 TeV αλ mixing angle as defined below Eq. (3). Using the above expressions, we can compute the Ω 2 where DMh represents the experimentally observed approximate dark matter abundance in the specific cases value. We can use these to infer approximate combinations where individual annihilation channels dominate the of masses and couplings that reproduce the observed dark freeze-out process: matter relic density. Excepting the case where the inter- 0 2 2 2 mediate Z is nearly on resonance, requiring perturbativity mψ 0.05 η 0 − 4 2 2 Z ψ ≲ 2 − 3 ðΩh Þχ0 χ0 ≈ Ω h ; ð11Þ up to the GUT scale imposes the bound m TeV. 1 1 DM 1 α0 2 TeV Somewhat higher masses are possible in regions of param- eter space where multiple channels contribute to dark 5For the nonsupersymmetric case the importance of the Z0H0 matter annihilation. channel was discussed in [14,15], assuming both the Z0 and H0 We now turn to a numerical treatment that encompasses receive their mass from the Higgs mechanism. more general cases with multiple channels and contributions. 075019-4 SIMPLE HIDDEN SECTOR DARK MATTER PHYS. REV. D 102, 075019 (2020) Ω 2 FIG. 1. Left panel (a): the black dashed contours give the dark matter relic density in units of the observed abundance DMh ; the solid 0 0 contour corresponds to points that produce the observed DM abundance. The dense set of contours in the χ1χ1 region corresponds to 0 cases where the DM annihilates through the Z resonance, mψ ≃ mZ0 =2. Right panel (b): contours denote T ¼ 0 annihilation cross 0 ˜ sections hσvi0, relevant for indirect detection of dark matter. In both panels, we have fixed v ¼ 1.25 TeV, mS ¼ 2.5 TeV, ˜ 650 1 5 λ 0 25 ϕ 0 mT ¼ GeV, mB˜ 0 ¼ . TeV, Aλ ¼ . TeV, and CP ¼ . We use a combination of FEYNRULES [35] and MICROMEGAS have a significant impact on the relic density, since [36–38] with numerical diagonalization through ASperGe annihilation cross sections do not depend strongly on [39] to determine the relic abundance as well as the T ¼ 0 the Higgs mass in this region. cross sections relevant for indirect detection. Our results are In the right panel, we show dominant annihilation shown in Fig. 1 for a representative parameter set, and ϵ channels at T ¼ 0, along with (dashed) contours of the sufficiently small that all SM final states can be neglected. annihilation cross section. The solid contours denote The color coding represents the strongest annihilation regions with the correct relic density. Again, over most channel at each point of parameter space. The left panel of the parameter space, annihilations to Z0H0 dominate. 0 0 0 0 shows that the χ1χ1 (blue) and Z H (gold) channels tend to Along the upper solid contour, the dominant p-wave dominate on either side of the fixed point η ¼ 1.Asα0 contribution (to H0H0) contributes a maximum of ∼3% increases, the lighter scalar mass decreases, owing to to the total annihilation cross section in the early Universe. σ the presence of the D-term, see Eq. (7). The right edge The result is that h viT¼0 is very nearly the s-wave value of 2 10−26 3 of the plot denotes mS1 ¼ mψ , beyond which the scalar is × cm =s along this contour. the LZP and the DM candidate. Close to this boundary, ψ In either panel, annihilation rates to neutralino final 0 0 and S1 are approximately degenerate, and coannihilation states do not exceed those to Z H when the latter channel processes can dominate the freeze-out process (green is kinematically unsuppressed. Relative to the Z0H0 final 2 region). The green region, corresponding to coannihilation state, whose tree-level cross section is ∝ αλ, s-channel 0 0 into χ1Z , features a resonant effect where the heavier contributions to neutralino final states are suppressed by 0 α0 α 1 ≡ neutralino χ2 can go approximately on-shell; this occurs powers of = λ or =xf, with xf mDM=Tfo. However, ≈ 2 ∼ 0 when mS1 þ mψ mψ mχ2 . In the upper-left corner of neutralino final states are still relevant, if subdominant, this plot, we expect relatively large loop corrections to the where the scalar exchange diagrams are sufficiently large. H0 mass, which we have not included in our relic density This occurs near the right side of the plot, where a sizable calculations. However, in this same region the hidden D-term acts to suppress one of the scalar masses. 2 sector would also be fine-tuned for this set of supersym- Incidentally, the leading s-wave αλ piece of neutralino metry breaking parameters, since the relatively light Z0 diagrams is suppressed by the scalar mass splitting, ∼λ2 ˜ 2 m2 − m2 .Thes-wave annihilation rate to neutralinos would receive substantial corrections going like mS.In S1 S2 0 addition, we do not expect the change in Higgs mass to assuming α ≪ αλ is 075019-5 BARNES, JOHNSON, PIERCE, and SHAKYA PHYS. REV. D 102, 075019 (2020) 2 4 πα mψ complicated pictures, including coannihilations or annihi- σ 2 − δ λ 2 − 2 h viχ0 χ0 ¼ CiCjð ijÞ 2 4 ð s2θ Þ i j 4 S lations to hidden neutralinos, are possible. mψ mS1 ð1 − m2 =m2 Þ2 S1 S2 IV. SCALAR DARK MATTER × 1 2 2 2 1 2 2 2 ; ð13Þ ð þ mψ =mS1 Þ ð þ mψ =mS2 Þ In this section, we consider the scenario where the lighter 2 2 scalar S1 is the LZP dark matter candidate. For S1 to be the where C1 ¼ cos θN, C2 ¼ sin θN. This additional sup- pression can be understood by taking the m2 → m2 limit LZP, one of two conditions are required: (i) one of the soft S1 S2 masses m˜ 2, m˜ 2 should be negative, or (ii) one of m˜ 2, m˜ 2 andperformingaFierztransformationonthesumofthe S T S T should not be too large (in this limit, SUSY relations force scalar mediated diagrams. In this limit, these diagrams 5 μ 5 the scalar to be degenerate with the ψ), and either the sum to give the operator ðψγ¯ μγ ψÞχ¯iγ γ χj,whichis trilinear term Aλ or D-term contribution must be large α ≫ α0 helicity suppressed [40]. In the limit λ , the cross enough to sufficiently split the eigenvalues to push the section for annihilation to neutralinos can reach exactly smaller eigenvalue below mψ. one-half the cross section of Z0H0, a limit saturated as For case (i) with a single negative soft mass, in order to m =m → ∞, m → mψ ,andS1 → T or S.Forthe S2 S1 S1 ensure open annihilation channels in the hidden sector, explicit case shown in Fig. 1,wherem =m ≲ 6, over a S2 S1 which is necessary to obtain the correct relic density via majority of the gold region annihilation to neutralinos freeze-out (since we assume ϵ to be small), we must further – 0 contributes roughly 10% 40% to the overall annihilation 2α α 0 have < λ or a seesawed down mχ1 . In the limit of rate, with smooth interpolation to 1 at the fixed point line. 0 αλ ≫ α the cross sections are well approximated by Figure 1 is only one slice of parameter space, and it is of interest to explore the dependence on other parameters πα2 2 2 ˜ 0 λ jAλj (mS;T, v , m ˜ 0 , and Aλ). The arguments of the previous hσvi → 0 0 ¼ 1 − ; ð14Þ B S1S1 Z Z 4m2 m2 þ m2 paragraph summarize the dominant effect of varying the S1 S1 S2 scalar soft masses—they act as a dial that changes the σv 0 0 relative importance of the neutralino final state(s) when h iS1S1→H H 0 0 2 2 2 2 2 both Z H and neutralino final state(s) are kinematically πα 2θ 1 2mψ − Aλ sin2θ λ 1 − jAλj cos S − j j S accessible. The m ˜0 chosen in the figure is such that the ¼ ; B 4 2 2 2 2 m 0 0 mS1 mS1 þ mS2 S1 χ1χ1 final state can go on resonance, yet small enough that the heavier neutralino is still accessible. A smaller mB˜ 0 ð15Þ would alleviate some of the heavy neutralino kinematic 2πα2 2 2 suppression, allowing a marginally lighter thermal dark 2 λ mS1 =mψ σ 0 0 2−δ 2θ h viS1S →χ χ ¼CiCjð ijÞsin S 2 2 2 2 ; matter in regions where annihilations to the heavy neu- 1 i j m ð1þm =mψ Þ tralinos are relevant. Recall that for the dark matter to be a S1 S1 fermion, the trilinear Aλ term must be small enough to not ð16Þ push a scalar mass below mψ. Otherwise, the primary effect 2θ 2θ of Aλ is reflected via the impact of the scalar masses on where Ci ¼ cos N, C2 ¼ sin N. Note that because the 0 0 annihilations to hidden neutralinos as described above. initial state is CP even, the relevant states are Z Z and 0 0 0 0 Finally, the relic density is controlled by the overall H H , in contrast to the fermion case where Z H played a 0 0 mass scale. In cases where an s-wave process dominates starring role. Of these states, H H typically dominates ≳ (typically the case here), in the freeze-out approximation, when mψ mS1 on account of the term that goes like 2 2 2 Ω ∼ m logðmÞ,wherem is the mass scale associated with mψ =mS1 in Eq. (15). This term is due to t- and u- channel the annihilation cross section, σ ∝ m−2.Itistherefore diagrams generated via the jTj2jH0j2 term with one H0 set to possible to shift any given contour to the correct its vev. Finally, the presence of sin 2θS in Eq. (16) can be DM abundance by rescaling (within the limits permitted understood by noting that this channel receives a helicity by perturbativity considerations) all the mass scales suppression in the absence of scalar mixing. in the hidden sector by the square root of the number Figure 2 shows the relative importance of these channels displayed. When coannihilations become important, the and illustrates a case where the correct relic abundance is 2 scaling is still roughly Ω ∝ m , but with corrections that realized via annihilation to neutralinos. In the figure, mψ ¼ 848 – cause some deviation from this behavior (see [41] for GeV and mS1 varies in the range 540 620 GeV. Even further details). with this relatively modest hierarchy, scalars still domi- In summary, we have shown that fermionic dark matter nantly annihilate to H0H0 when this channel is kine- with simple thermal histories is possible in our framework. matically accessible. In this case, and for relatively In the majority of the parameter space, annihilations to small A-terms, the dark matter abundance may be approxi- Z0H0 provide for the “hidden WIMP miracle,” but more mated as 075019-6 SIMPLE HIDDEN SECTOR DARK MATTER PHYS. REV. D 102, 075019 (2020) FIG. 2. Left panel (a): relative contributions of various annihilation processes to S1 freeze-out. The relative importance is given by the σ h vii Micromegas output fi ≡ σ and corresponds to a freeze-out approximation for the annihilation rate of each channel [38]. Right panel h vitotal 0 (b): the annihilation cross section for T ¼ 0, relevant for indirect detection. For both panels, αλ ¼ 0.045, v ¼ 1.6 TeV, ˜ 2 −4002 2 ˜ 1500 3000 λ 600 ϕ −π mS ¼ GeV , mT ¼ GeV, MB˜ 0 ¼ GeV, Aλ ¼ GeV, and CP ¼ . 0.12 4 2 TeV 4 m 6 the correct DM abundance with this channel (even away Ω 2 ≈ Ω 2 S1 ð S1 h ÞH0H0 DMh : 0 0 λ mψ 500 GeV from the Z , H pole). For case (ii), with positive soft masses, an interesting ð17Þ feature (for relatively modest mB˜ 0 ) is a relatively com- pressed spectrum, with the gauge boson, Higgs boson, scalars, and fermions all in close proximity. The Aλ term is However, for large α0, this channel becomes kinematically not expected to be too large compared to the scalar masses, inaccessible, and the relic density is set by annihilation into hence relatively compressed spectra are quite generic 0 0 the only available channel in the hidden sector, χ1χ1. unless the D-term is very large. Such compressed spectra Although this channel is suppressed relative to the bosonic result in potentially richer cosmologies, including more final states, Fig. 2 shows that it is still possible to achieve robust possibilities of coannihilation. FIG. 3. Left panel (a): color coding represents the dominant annihilation process contributing to S1 freeze-out, while contours represent the computed DM relic abundance in units of the observed relic density. Right panel (b): the annihilation cross section for T ¼ 0, relevant for indirect detection. Again, different colors indicate the dominant annihilation channel. For both panels, α0 ¼ 0.01, 0 2 ˜ 400 400 λ 200 ϕ 0 v ¼ TeV, mT ¼ GeV, MB˜ 0 ¼ GeV, Aλ ¼ GeV, and CP ¼ . 075019-7 BARNES, JOHNSON, PIERCE, and SHAKYA PHYS. REV. D 102, 075019 (2020) The interplay of the above processes can be seen in density. Moreover, the small mass gap can cause the heavier Fig. 3, where we plot the dominant annihilation processes of the two to have a long lifetime, which can have important over a slice of parameter space. At low αλ, the only cosmological and phenomenological consequences. † kinematically accessible state for S1S1 annihilation is The heavier state decays to its superpartner and a trio of 0 0 χ1χ1 (blue region). However, in this blue region and for SM fermions via an off-shell neutralino and the RPV αλ ≳ 0.02, there is also an open coannihilation channel coupling through a dimension-7 operator, which we write 0 0 (Z χ1). This coannihilation is exponentially suppressed schematically as because the S1—ψ mass splitting is still substantial, ψ ψ ψ ψ ∼Oð100Þ GeV; however, for a slice in the bottom left O ðS1 Þð SM SM SMÞ decay ¼ 3 ; ð18Þ (red), a near perfect destructive interference among Λ diagrams contributing to the χ0 χ0 annihilation nevertheless 1 1 where Λ is a combination of gaugino and sfermion masses, allows for coannihilation to dominate. However, because ψ the overall annihilation rate is exceedingly small, the and SM represents a SM fermion. The identity of the relic density far exceeds the observed relic density. For fermions depends on the texture of RPV couplings but does not affect our discussion here. The important effect is that larger αλ, annihilations to dark Higgs and gauge bosons 0 the large power of Λ in the denominator, coupled with the dominate as they become accessible at αλ > 2α (gold region). Eventually, the χ0 χ0 (green region) state becomes phase space suppression due to the small mass splitting 1 2 and the 4-body final state, can lead to an extremely long kinematically accessible and marginally exceeds these decay lifetime. Assuming ψ to be lighter without loss of channels. The neutralino channels diminish as we move generality, the decay width for S1 → ψψ ψ ψ is to the right side of either panel, owing to the decrease in SM SM SM schematically sin 2θS; see the discussion surrounding Eq. (16).For sufficiently small θ , the process is driven by otherwise S ϵ2g02g2 λ002 Δm7 subdominant pieces suppressed by g0=λ, which are not Γ ∼ Y ; 19 29π5 4 2 ð Þ shown in Eq. (16). Note that, again, the largest values of αλ m0mLSP shown here correspond to tuned hidden sectors, particularly Δ − where m ¼ mS1 mψ , and m0 represents a generic scalar for the largest values of mS. The fact that Z0Z0 exceeds H0H0 here is a consequence of superpartner mass in the visible sector. This decay width the relatively degenerate spectrum; the enhancement that corresponds to a lifetime hidden Higgs final states receive from factors of mψ =m , S1 −4 2 7 4 2 10 10 GeV m0 m discussed below Eq. (15), is no longer substantial. For τ=s ∼ LSP : ð20Þ example, in the bottom right corner when Z0Z0 goes on- ϵg0λ00 Δm TeV TeV 0 shell, mψ =m ∼ 1.09. This permits terms proportional to g S1 τ ≳ 1 [neglected in Eq. (15)] to allow annihilation to Z0Z0 to A lifetime s could disrupt big bang nucleosynthesis (BBN) through late injection of energetic photons and dominate. The S1 and ψ grow closer in mass toward the top right, and coannihilation processes are seen to become charged fermions. This potentially imposes a strong con- Δ important (red region). As discussed in the previous straint on m and hence coannihilation as a viable method section, any of the contours can be made to match the to produce the correct dark matter relic density, though the correct relic density by rescaling the mass scales involved. strength of the constraint depends on other unknown We see from these figures that unlike when ψ is the LZP, parameters. One possibility that allows for sufficiently — Δm— the case where S1 is the LZP is more involved, with several short lifetimes even in the presence of small is ϵ available annihilation processes that are viable candidates for to be fairly large, which would have interesting for setting the relic abundance. The link between indirect implications for direct detection. detection and freeze out, shown by comparing the left and Another possibility, following [42,43], is to note that right panels of Figs. 2 and 3, is, however, relatively particle decays after BBN are allowed so long as the energy straightforward due to the nearly universal presence of injected into photons and e is several orders of magnitude s-wave processes. Exceptions occur in regions where smaller than the energy density in dark matter. This can coannhilation processes dominate, rendering a suppressed indeed be the case. Understanding the constraints requires indirect detection signal. an understanding of the energy density stored in S1.For instance, suppose the coannihilation process that sets the ψ 0 0 dark matter relic abundance is ψS1 → χ1Z . In this case, the V. COANNIHILATION REGIME 0 0 0 0 cross-processes S1Z → ψχ1 and S1χ1 → ψZ can continue The regime where the mass gap between the LZP and its to deplete the S1 abundance until they freeze out at a later ≈ superpartner is small is worthy of special attention. As seen time. Assuming mψ mS1 , we can approximate the energy in earlier sections, in this regime coannihilations between density in S1 relative to the energy density in ψ DM after S1 the two can be important for setting the dark matter relic freeze-out as 075019-8 SIMPLE HIDDEN SECTOR DARK MATTER PHYS. REV. D 102, 075019 (2020) ρ mψ mψ whereas the lower (upper) boundary for the range of the S1 ∼ − − 1 Exp 0 ; ð21Þ 2 α0 ρψ T minðmχ0 ;ZÞ m ˜ 0 coefficient corresponds to small (large) .Justasthe fo 1 B0 large top Yukawa in the MSSM suppresses stop masses in where Tfo is the DM freeze-out temperature. Due to this the IR, large λ can suppress the dark scalar masses in the IR. exponential dependence, we expect lifetimes with post- And as in the (no-scale) MSSM, the gaugino mass can mψ BBN decays to be compatible for 0 ≳ 1.4 even if all minðmχ0 ;Z Þ generate scalar masses at one-loop. However, because the 1 Abelian dark Uð1Þ0 runs to weak coupling in the IR, this of the S1 energy density were ultimately converted into suppresses the IR gaugino mass and mitigates its effects on photons and e . Note, however, that much of the S1 energy the scalar masses. goes into LZP dark matter, and only a small fraction To understand what we expect for the hidden sector mass ∼Δm=m goes into photons and e , significantly mitigat- S1 scale in Eq. (22), we should compare to the parameters of ing such constraints. Furthermore, given that n ≪ n at S1 DM the MSSM, whose values we have some indirect clues the time of dark matter freeze-out, we also do not expect about from the LHC. We make use of the approximate subsequent S1 scattering or decay processes to contribute a solutions to MSSM RGEs from, e.g., [44–47].For significant additional population of dark matter. tan β ¼ 10, IR SUSY breaking parameters are related to universal boundary conditions for scalars (m0Þ, gauginos VI. UV CONSIDERATIONS (m1=2), and trilinears (A0) as: In the previous sections, we illustrated several incarna- 2 2 2 m˜ ≈ 0.63m þ 5.7m − 0.13m1 2A0; ð24Þ tions of the WIMP miracle in the hidden sector that differed Q3 0 1=2 = both in the identity of the dark matter and the most 2 2 2 m˜ ≈ 0.26m þ 4.3m − 0.27m1 2A0; ð25Þ important annihilation channel. In this section, we examine U3 0 1=2 = whether and how these various scenarios arise from ˜ 2 ≈−0 12 2 − 2 6 2 − 0 40 reasonable choices of parameters at a high scale, and mH2 . m0 . m1=2 . m1=2A0; ð26Þ whether these are compatible with constraints on weak scale MSSM parameters. ðM3;M2;M1Þ ≈ ð2.9; 0.82; 0.41Þm1=2: ð27Þ We pay particular attention to what mass scales are reasonable in the hidden sector under the assumption that We are agnostic about the precise UV SUSY-breaking supersymmetry breaking is communicated similarly to the boundary conditions, but universal boundary conditions two sectors, as might occur with gravity mediation. One such as these enable us to get a sense of the rough scales must renormalization group (RG) evolve the resulting involved. parameters from the scale at which SUSY breaking is Some of these parameters are constrained by LHC data. mediated to the weak scale, relevant for dark matter In particular, direct searches constrain stops and gluinos to phenomenology. Evolution of the hidden sector parameters be TeV scale or heavier. But it is also possible to say more. is performed with β-functions from Ref. [32] with slight Given the absence of a definitive hint of the mass scale of corrections (see Appendix for details). A linear combina- superpartners, we can take the measured mass of the Higgs 125 tion of soft masses convenient for RG evolution, which also boson, mh ¼ GeV, as an indirect measure of the stop serves as a rough proxy for the overall scale of the hidden mass scale. It is known that this Higgs mass is compatible sector, is with stop masses below a TeV in the presence of significant stop mixing, but LHC direct searches place this scenario in 1 2 2 2 tension. If mixing in the stop sector is not near maximal, Σ ≡ m˜ m˜ m˜ 0 : 22 3 ð T þ S þ H Þ ð Þ then the observed Higgs mass requires that stops be much heavier, ≳5 TeV [48–51]. From the equations above, we The physical mass spectrum is related through see that the stop mass in the IR is largely determined by two 2 2 2 2 2 2Σ 2 ðmS1 þmS2 Þ= ¼TrðmscalarÞ= ¼3 þmψ .Whenevolved UV mass parameters: a soft mass scale m0 and the gaugino into the IR, which for concreteness we evaluate at the mass scale m1=2. The gaugino mass piece provides top quark mass, we find the following approximate the dominant contribution in the IR unless m0 ≫ m1=2.In expression, derived from numerical RGE flow graphs the gaugino mass dominated scenario, ∼5 TeV stops sug- shown in the Appendix, Fig. 10: gest m1 2;m0 ∼ TeV. Assuming that Σ0, m ˜ 0 are compa- = B0 2 rable to their respective MSSM counterparts m and m1 2 in Σ 0 3 1 Σ 0 0 4 2 0 = ðmtopÞ¼ð . ; Þ 0 þð ; . Þm ˜ 0 : ð23Þ B0 the UV, hidden sector RG running then suggests Oð100Þ GeV–OðTeVÞ as the mass scale for hidden sector Here m ˜ 0 is the value of the soft mass of the hidden gaugino B0 particles. Scalar mass dominated scenarios are correlated at the high scale. The lower (upper) boundary of the range with somewhat heavier hidden sector masses. Therefore, of the Σ0 coefficient corresponds to large (small) αλ, Oð100Þ GeV–OðTeVÞ scale hidden sector particles can be 075019-9 BARNES, JOHNSON, PIERCE, and SHAKYA PHYS. REV. D 102, 075019 (2020) green). When the mass splitting is small, coannihilations can dominate (dot-dashed curve). This figure therefore shows that for a range of TeV scale input parameters in the UV, the correct relic abundance can be realized for both fermion and scalar dark matter scenarios. It is worth pointing out that at the right edge of the plot, S1 transitions from being ˜ ˜ dominantly T to S. For sufficiently large m˜ T, m˜ S will be pushed down via the impact of m˜ T on the RG flow of m˜ S,so further increase of m˜ T actually results in a decrease in the lightest scalar mass. Finally, we comment briefly on the evolutionpffiffiffi of the 0 FIG. 4. Interpolation between dark matter scenarios by scan- dimensionless couplings. There is a λ ¼ 2g IR fixed ˜ 0 ning the soft mass mT in the UV. The solid black curve indicates point, which can be understood as the emergence of an the realized dark matter density, while the other curves indicate N ¼ 2 SUSY, where ψ is degenerate with Z0, H0. The the relative importance of different annihilation channels, with fi distance from this fixed point, r, has a simple solution at as defined in the caption of Fig. 2. The parameters in the legend one loop are defined in the UV. 0 0 3 2α ðtÞ α ðtÞ r0 ≡ 1 − 0 rðtÞ ;rðtÞ¼r 0 ¼ α0 t 3 ; ð28Þ αλðtÞ α0 0 generically compatible with multi-TeV stops and gluinos in ð1 − π Þ the MSSM sector without any significant mass hierarchies in ≡ μ 2 1016 the UV. Detailed information on the hidden sector spectrum where t logð =MGUTÞ, MGUT ¼ × GeV, and null requires RG evolution of the splittings between the various subscripts correspond to GUT boundary conditions. soft masses. The splittings decrease fairly slowly and so are Interestingly, rðtÞ is also a measure of the kinematic sensitive to their UV starting points. suppression of ψψ¯ annihilation to Z0, H0 (see Fig. 9 and We now turn from these generalities to make a more firm surrounding text in the Appendix). This equation thus connection with the cosmological histories outlined in indicates the possibility of kinematic suppression of this earlier sections. We use analytic one-loop formulas for channel as a consequence of RG evolution. IR quantities (see Appendix) to plot relic abundances given UV initial conditions. In Fig. 4, we show (as a black curve) VII. DECAY MODES OF HIDDEN SECTOR the realized relic abundance as a function of the IR mass PARTICLES splitting between the lightest scalar S1 and the fermion ψ ˜ 0 In this section we discuss the decay modes of various in the hidden sector. This splitting depends on mT in the UV. The various curves denote the relative importance hidden sector particles. This is crucial for indirect detection, of various annihilation channels—solid (dotted) curves as once the dark matter annihilates into these particles, their for fermion (scalar) dark matter—as well as coannihila- decay modes will determine the spectra of SM states that − will be observed by experiments. tion (dashed curves). At large mS1 mψ , we can see that the correct relic abundance is realized for a wide range A. H0 decays of mass splittings, consistent with thermal histories domi- nated by ψψ¯ → Z0H0 annihilations (solid orange curve). Kinetic mixing induces a mixed quartic interaction Annihilations to the lighter neutralino are smaller by Oð1Þ between the visible and hidden sector Higgs fields via a factors and therefore not negligible (solid green curve). On D-term contribution to the potential shown in Eq. (5). This the left hand edge of the plot, where scalar dark matter is generates a mixed mass matrix after each field acquires a → χ0 χ0 0 0 0 ϵ realized, S1S1 1 1 annihilation dominates (dotted vev. In the ðHd;Hu;HÞ basis, this is (to leading order in ) 0 1 2 2 2 2 2 2 sβm þ cβm −sβcβðm þ m Þ ϵsθ m m 0 cβ B A Z A Z W Z H C 2 B − 2 2 2 2 2 2 δ 2 −ϵ C m 0 ¼ sβcβðm þ m Þ cβm þ sβm þ =sβ sθ m m 0 sβ ; ð29Þ Hd;Hu;H @ A Z A Z W Z H A 2 ϵsθ m m 0 cβ −ϵsθ m m 0 sβ m W Z H W Z H H0 where we have used the abbreviations sx ¼sinx and cx ¼ we adjust to recover the 125 GeV Higgs mass. After cos x, and tan β ¼ vu is the ratio of the up- and down-type rotating the visible sector Higgs fields by the standard vd 2 2 2 246 2 δ MSSM Higgs mixing angle α, the mass matrix in the Higgs vevs, with vu þ vd ¼ v ¼ð GeVÞ . The term encodes the radiative contribution to the Higgs mass, which ðH; h; H0Þ basis is 075019-10 SIMPLE HIDDEN SECTOR DARK MATTER PHYS. REV. D 102, 075019 (2020) 0 1 2 m 0 ϵsθ m m 0 cα β B H W Z H þ C 2 B 2 C 0 m −ϵsθ m m 0 sα β mH;h;H0 ¼ @ h W Z H þ A: ð30Þ 2 ϵsθ m m 0 cα β −ϵsθ m m 0 sα β m W Z H þ W Z H þ H0 π ≫ α ≈ β − ˜ In the MSSM decoupling limit, mA mZ, 2.IfH supersymmetric limit, recall that a somewhat large mB0 6 ˜ 0 0 is so massive that it decouples from this system, produces a seesaw effect that makes (the mostly H ) χ1 we get 0 0 0 light, opening the channel H → χ1χ1. The width for this ! 2 channel is m ϵsθ m m 0 c2β 2 h W Z H m 0 ¼ : ð31Þ 2 3 2 h;H 2 02 4 = ϵsθ m m 0 c2β m g m 0 mχ0 W Z H H0 Γ 0 → χ0 χ0 H 2 θ 2 θ 1 − 1 ðH 1 1Þ¼ 4π ðsin N cos NÞ 2 ; mH0 The mass eigenstates of this matrix are comprised 0 ð33Þ of the h, H states with mixing angle θH approximately given by ˜ 0 ˜ 0 where θN is the B − H mixing angle, see Eq. (3). ϵsθ m m 0 c2β If this decay channel is not kinematically accessible, the θ ≈− W Z H H 2 2 : ð32Þ 0 ϵ2 m 0 − m H decays into SM states with an suppression. Because H h 0 H inherits the couplings of the SM Higgs via θH mixing, it To understand which decays are allowed for H0 may decay into SM states such as WW, ZZ, tt¯ or hh [52],or requires an understanding of the H0 mass relative to to visible sector superpartners, especially neutralinos. those of other hidden sector particles. As discussed in Decays to hh are also directly mediated via the Higgs Sec. II, we expect an approximate degeneracy between portal coupling. Decays to MSSM Higgs states, e.g., AA, − H0 and Z0 to be maintained even after accounting for HH, HþH , are possible but likely kinematically sup- loop corrections. This eliminates the possibility of the pressed, and we do not consider them further for simplicity. decay channel H0 → Z0Z0, except in extremely fine- The final possibility is the decay into neutralinos of both 0 0 tuned regions (for moderate fine-tuning, it might be sectors, H → χ1χ1, which occurs at the same order in ϵ. possible that ZZ0 could be open). More likely is the Among the visible sector SM states, H0 → WW will 0 possibility of H decays to neutralinos. While the hidden dominate so long as mH0 > 2mW. In the approximation sector neutralinos are also degenerate with the H0 in the given by Eq. (32), the partial widths to SM bosons are: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϵ2 2 2 2 4 2 g c2βmH0 1 − 4ðm =m 0 Þ þ 12ðm =m 0 Þ 4m Γ H0 → WW Y W H W H 1 − W; ð Þ¼ 64π 1 − 2 2 2 mh=mH0 mH0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϵ2 2 2 2 4 2 g c2βmH0 1 − 4ðm =m 0 Þ þ 12ðm =m 0 Þ 4m Γ H0 → ZZ Y Z H Z H 1 − Z; ð Þ¼ 128π 1 − 2 2 2 mh=mH0 mH0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 2 ϵ g c2βm 0 3c2βm 4m Γ 0 → Y H 1 Z 1 − h ðH hhÞ¼ 128π þ 2 − 2 2 : ð34Þ mH0 mh mH0 0 0 0 In the large mH0 limit, we get ΓðH →WWÞ≈2ΓðH →ZZÞ ≈ The decay width into MSSM neutralinos H → χiχj is 2ΓðH0→hhÞ as expected from the Goldstone equivalence subdominant to the above widths due to the relatively small mZ theorem. Yukawa coupling suppressed by μ . The decay into the 0 0 neutralino combination H → χ1χ1, on the other hand, can 6 dominate in some regions of parameter space. This process One must take care in taking this strict decoupling limit. For is generated by neutralino mixing. After diagonalizing the 0 → χ χ processes such as the decay H 1 1, the contribution via Hu kinetic terms of the neutralinos via B˜ → B˜ − ϵB˜ 0, the mass may be suppressed relative to those from H , for instance due to d matrix in the basis χ˜0 ¼ðH˜ 0B˜ 0jB˜ H˜ H˜ Þ is (to leading χ1 having a roughly tan β larger content of Hd ∼ H than Hu ∼ h. gauge D U In this case, effective decoupling can be delayed. order in ϵ) 075019-11 BARNES, JOHNSON, PIERCE, and SHAKYA PHYS. REV. D 102, 075019 (2020) 0 0 FIG. 5. Branching ratio of the hidden Higgs boson to the lightest MSSM and hidden neutralino, BRðH → χ1χ1Þ, as contours in the ˜ — ˜ ˜ ˜ 600 −600 hidden gaugino mass mB0 mixed gaugino mass mBB0 plane (left panel) and for mB0 ¼ , GeV (right panel). The other β 10 150 μ 1000 ϵ 01 0 1 500 parameters are set to tan ¼ , mB˜ ¼ GeV, ¼ GeV, ¼ : , g ¼ , and mH0 ¼ GeV. 0 1 0 g0v0 000 B C B 0 0 1 1 C B g v m ˜ 0 ϵm ˜ ˜ 0 − ϵm ˜ g ϵv − g ϵv C B B BB B 2 Y d 2 Y u C B 1 1 C B 0 ϵm ˜ ˜ 0 − ϵm ˜ m ˜ − g v g v C mχ ¼ B BB B B 2 Y d 2 Y u C: ð35Þ B C B 0 1 g ϵv − 1 g v 0 −μ C @ 2 Y d 2 Y d A 1 1 0 − 2 gYϵvu 2 gYvu −μ 0 Here, we assume that the wino is sufficiently heavy ratio into this decay channel in the H decoupling limit to be decoupled from the analysis. Diagonalizing the and to leading order in ϵ. In the contour plot in the † ˜ neutralino and Higgs mass matrices as mχ;diag ¼ NmχN , left panel, the branching ratio vanishes for small jmB0 j 2 2 † m ¼ Um 0 U , we calculate the relevant decay because the channel becomes kinematically inaccessible, diag Hd;Hu;H 0 0 ˜ width as mχ þ mχ >mH . At large jmB0 j it is negligible because the 0 0 0 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hidden sector decay H → χ1χ1 becomes kinematically u 2 2 2 2 02 u 2 4 accessible and dominates. In between, large mixing g m 0 t mχ mχ0 mχ1 mχ0 Γ 0 → χ χ0 H 1− 1 − 1 − 1 between the hidden neutralino and the lightest MSSM ðH 1 1Þ¼ 8π 2 2 4 mH0 mH0 mH0 neutralino can make this mixed channel dominant, reaching 2 2 branching ratios over 90%. The right panel shows two mχ þm 0 þ2mχ mχ0 1 χ1 1 1 1− slices of this contour plot at m ˜0 ¼ 600, −600 GeV. This × 2 B mH0 plot illustrates that the relevant branching ratio can vanish 2 for some value of m ˜ ˜ 0 where contributions from field ×ðU10;H0 ðN1;B˜ 0 N10;H˜ 0 þN1;H˜ 0 N10;B˜ 0 ÞÞ : ð36Þ BB redefinition to remove the kinetic mixing of Eq. (2) and the The neutralino masses are allowed to be negative in diagonalization to remove the mass mixing introduced in ˜ this formula, and the first index of Nij, Uij denotes Eq. (4) conspire to cancel each other. For χ1 ∼ B (equiv- 0 0 mass eigenstates, with i ¼ 1 ; 2 ; 1; 2; 3, with the alently, mZ ≪ μ) and small ϵ, this cancellation occurs when ≈ prime indicating that the eigenstate is dominantly com- mB˜ mB˜ B˜ 0 , as seen from the relevant off-diagonal term prised of hidden sector fields. The second index indicates in Eq. (35). states after diagonalizing kinetic terms but prior to mass In summary, we expect H0 decays to be dominated by 0 0 0 0 0 0 0 0 diagonalization. In Fig. 5, we explore the branching H → WW, H → χ1χ1, H → χ1χ1,orH → Z Z. Decays 075019-12 SIMPLE HIDDEN SECTOR DARK MATTER PHYS. REV. D 102, 075019 (2020) the visible sector via the portal coupling and are ϵ sup- pressed. The decay must proceed through the gaugino 0 component of χ1, denoted by N101. In the limit where R- 0 parity is unbroken, χ1 decays into the LSP χ1 via an off- shell sfermion, with the width [22,53] 2 2 2 5 ϵ α N mχ0 Γ χ0 → χ ¯ Y 101 1 2 2 ð 1 1ffÞ¼ 4 f2ðmχ1 =mχ0 Þ; ð37Þ 64π m0 1 3 4 2 where f2ðxÞ¼1 − 8x þ 8x − x − 12x log x, and m0 is the sfermion mass scale. If kinematically allowed, it can 0 also decay as χ1 → χ1ðh=ZÞ through the bino-Higgsino 0 mixing in the visible sector (if the χ1 − χ1 splitting is FIG. 6. Branching ratio of the hidden gauge boson to the smaller than the h=Z mass, the boson can be off-shell, 0 0 giving a 3-body decay). This on-shell decay channel is lightest MSSM and hidden neutralino, BRðZ → χ1χ1Þ, for 8π m0 2 ˜ 600 −600 subdominant to the above channel if jμj > ð Þ mZ.If mB0 ¼ , GeV. The other parameters are set as in Fig. 5: gY mχ0 0 1 tan β ¼ 10, m ˜ ¼ 150 GeV, μ ¼ 1000 GeV, ϵ ¼ :01, g ¼ 1, 0 B R-parity violation is significant, χ1 inherits the RPV decay and mH0 ¼ 500 GeV. channel of χ1 into three SM fermions: 5 2 002 2 m to hidden sector particles will be followed by cascades into 3ϵ λ N 0 α χ0 Γ χ0 → ¯ ¯ ¯ 1 1 Y 1 ð 1 udd þ u d dÞ¼ 2 4 : ð38Þ SM final states. 128π m0 B. Z0 decays For sufficiently large λ00, this can be the dominant decay χ0 0 0 0 channel for 1. The hidden sector decay Z → χ1χ1 will dominate if ϵ kinematically accessible since all other channels are VIII. DIRECT DETECTION AND COLLIDER suppressed; otherwise, decays to pairs of SM fermions 0 CONSTRAINTS or to χ1χ1 dominate. In the mZ0 ≫ mZ limit, the couplings of the Z0 to fermions are simply proportional to hyper- Direct detection in scenarios of hidden sector dark matter charge as induced by the kinetic mixing, and the residual through the kinetic mixing portal has previously been change to the coupling coming from the diagonalization of studied by, e.g., [16,54]. The spin-independent cross the Z − Z0 mass matrix is negligible. The Z0 dominantly section per nucleon can be written to leading order in ϵ as: decays to up-type quarks, followed closely by charged 2 2 2 2 2 2 0 ϵ 0 μ leptons. In the opposite m 0 ≪ m limit, the Z instead C g e Z cθ Dn Z Z σ ¼ W ; ð39Þ primarily couples to electric charge, again decaying domi- π 2 4 A mZ0 nantly to up-type quarks (except the top quark, which is μ now kinematically inaccessible) or charged leptons. Decays where Dn is the reduced mass of the dark matter particle to WW or hZ are small, at the percent level or below. and the nucleon, Z is the atomic number, A is the mass 0 0 1 4 θ For Z → χ1χ1 decay to dominate requires a larger number, and C ¼f4 ; cos Sg for {fermion, scalar} dark neutralino mass mixing between the two sectors than in matter, with θS the mixing angle between S1 and S2. the Higgs boson case, see Fig. 6. This is because unlike Because mZ is much greater than the momentum exchange the H0, which couples to the gaugino-Higgsino com- in the scattering process, the cross section goes like bination in neutralinos, the Z0 couples to the Higgsino- coupling to the electromagnetic current e2Z2. 0 ˜ 0 Higgsino combination. Since the χ1 is mostly H and the Given these cross sections, we can derive constraints ˜ 0 0ϵ ϵ-suppressed coupling to χ1 is via the B component, the on the product g from direct detection experiments. 0 0 ˜ 0 ˜0 Current constraints from XENON1T [3] and projected Z → χ1χ1 coupling suffers from an additional H − B 0 0 constraints from LZ [55] are shown in Fig. 7. We plot mixing angle suppression relative to the H → χ1χ1 cou- pling. The relative closeness of the two curves for different two curves for each experiment, assuming mψ ¼ mZ0 or m 0 0 mψ ¼ Z for fermion dark matter. The choices are repre- signs of mB˜ compared to the H decay case (Fig. 5 right 2 panel) is merely an artifact of our choice of parameters, and sentative of different parameter regimes that replicate the θ correct relic abundance: The choice mψ ¼ m 0 is inspired can be modified by changing N. pffiffiffi Z 0 mZ0 by the IR fixed point λ ¼ 2g , whereas mψ ¼ 2 repre- 0 C. χ 1 decays sents the region of parameter space where the annihilation 0 ψψ¯ → χ0χ0 0 We are interested in scenarios where χ1 is the lightest i j occurs through a Z resonance (note that small mZ0 fermion in the hidden sector; hence all of its decays are into variations around the mψ ¼ 2 resonance can precisely 075019-13 BARNES, JOHNSON, PIERCE, and SHAKYA PHYS. REV. D 102, 075019 (2020) WIMP miracle, and annihilations into hidden sector states are unsuppressed by ϵ and typically dominated by s-wave processes. Thus, the present day dark matter annihilation cross sections can be large enough to make indirect detection a potentially powerful probe. Hidden sector decays are rapid enough to be considered prompt for indirect detection signals.7 The indirect detection signals will be sensitive to the mass spectra in both hidden and visible sectors, as well as to whether R-parity is conserved or broken. The possibil- ities are numerous. For now, we limit the discussion to qualitative comments, and leave a detailed treatment of various possibilities, including calculations of the precise FIG. 7. Constraints on the product g0ϵ from direct detection for spectra of SM final states and limits/projections from mZ0 fermion dark matter for mψ ¼ mZ0 (blue) and mψ ¼ 2 (green). various experiments such as Fermi and CTA [66], to future Solid lines denote constraints from XENON1T, and dotted lines work [67]. denote projected constraints from LZ after 1000 live days with a For fermionic dark matter, the dominant annihilation 5.6 tonne fiducial mass. LHC constraints are plotted assuming channel over the vast majority of the parameter space is 0 g ¼ 1, as derived from CMS searches for a resonance decaying ψψ¯ → Z0H0, followed by decay via portals to the visible 110 200 to muon pairs [56] for GeV 075019-14 SIMPLE HIDDEN SECTOR DARK MATTER PHYS. REV. D 102, 075019 (2020) spectrum is more compressed, the dominant annihilation matter annihilation into visible particles proceeds via a 0 0 channel over much of the parameter space is S1S1 → Z Z , series of cascade decays involving hidden sector particles, with the Z0 primarily decaying into SM fermions. This case and can lead to a wide variety of indirect detection signals also admits regions of parameter space where annihilation that might be within reach of future experiments; detailed to neutralinos (and their attendant cascades, as described studies of such signals will be performed in a future above) can be important, again leading to multiple SM paper [67]. fermions in the final state. The realistic hidden sector dark matter scenarios con- ACKNOWLEDGMENTS sidered in this paper can therefore lead to more complicated signatures compared to “simplified” hidden sector dark We thank Stefania Gori for helpful conversations. The matter scenarios (such as those considered in [14,15]), work of P. B., Z. J. and A. P. was supported by the DoE which generally consist of two-step dark matter annihila- under Grant No. DE-SC0007859. B. S. thanks the GGI tions of the form DM þ DM → Z0H0 → 4f. Institute for Theoretical Physics, the Mainz Institute for Theoretical Physics (MITP) of the Cluster of Excellence X. CONCLUSIONS PRISMA+ (Project ID 39083149), and the Leinweber Center for Theoretical Physics (LCTP) at the University In this paper, we have put together a simple framework of Michigan, where parts of this research were completed, for dark matter, building upon ingredients and guiding for hospitality and support. principles that are well-motivated: hidden sectors, super- symmetry, naturalness, and the realization of the correct APPENDIX: relic density for dark matter via the WIMP miracle. A hidden sector can lie around the weak scale, thereby We first present analytic one-loop solutions to the realizing the WIMP miracle. This happens naturally in RGEs of the model considered in this paper. We define ≡ μ ≃ 2 1016 scenarios where, for instance, supersymmetry breaking is t logð =MGUTÞ, with MGUT × GeV. UV boun- mediated to both the hidden and visible sectors via gravity dary conditions will be specified with a 0 subscript mediation. This can be made compatible with stringent at t ¼ 0. LHC limits on superpartners with only Oð1Þ differences between hidden and visible sector parameters in the UV. In α0 α0 0 this framework, we studied the minimal matter field content ðtÞ¼ 0 ; ðA1Þ 1 − α0t=π under a hidden sector Uð1Þ0 gauge symmetry that kineti- cally mixes with the SM hypercharge, where dark matter is −4π 0 α α F ðtÞ stabilized not by R-parity but by an accidental Z2 symmetry λðtÞ¼ λ0 ; ðA2Þ 1 þ 6αλ FðtÞ in the hidden sector. While the electroweak scale in our 0 sector might be accidentally small, we assumed symmetry α0 α02 2 “ ” − t 3 − 3 0t 0 t breaking in the hidden sector to be natural, which FðtÞ¼ 12π π þ π2 : ðA3Þ suggests that the hidden sector scalars and fermions as well as their superpartners lie around the same mass scale, 0 0 opening possibilities for a variety of dark matter candidates Note that since t< in the IR, FðtÞ > and increases as well as rich cosmological histories and indirect detection monotonically. In this Appendix, when we display quan- signatures. tities in the IR, for concreteness, we evaluate them μ For fermion dark matter, we found that dark matter at ¼ mtop. annihilation is generally dominated by the Z0H0 channel, The two-loop numerical RG flow is shown in Fig. 8, though annihilations to hidden sectors neutralinos are still and is well approximated by the above formulae. Recall 0 relevant. For scalar dark matter, annihilations to χ0χ0 , H0H0 that these couplings have a fixed point at 2α ¼ αλ, where i j ψ 0 0 as well as Z0Z0 states were shown to lead to consistent the becomes degenerate with the Z , H . We display the impact of this fixed point on the possible kinematic cosmological histories. We also found instances of coan- 0 0 nihilation between the scalar and the fermion providing the suppression of fermion annihilation to Z H in Fig. 9 (as correct relic density, where the heavier of the two can be discussed near Eq. (28)). For large couplings, it is extremely long-lived, well beyond BBN, yet consistent possible that evolution toward the fixed point can cause with all cosmological constraints. substantial suppression of the annihilation rate into the 0 0 In such frameworks, dark matter direct detection cross Z H final state. sections and production cross sections for hidden sector The coupling evolution itself is enough to determine two particles at colliders are generally suppressed by the portal more RG parameters: ϵ coupling strength mixing the two sectors. While such α0 ϵ ðtÞ signals might be observed, a too-small would preclude m ˜ 0 ðtÞ¼m ˜ 0 ; ðA4Þ B B0 α0 such possibilities. Indirect detection is different: dark 0 075019-15 BARNES, JOHNSON, PIERCE, and SHAKYA PHYS. REV. D 102, 075019 (2020) (a) (b) α0 FIG. 8. The IR couplings resulting from a two-loop numerical RG flow from MGUT to mtop. Left panel: The gauge coupling . Right panel: The Yukawa coupling αλ. 1 Δ ≡ 2m˜ 2 − m˜ 2 m˜ 2 ; AλðtÞ¼Aλ0 STH S ð T þ H0 Þ ðA7Þ 1 þ 6αλ FðtÞ 0 2 α0 0 − 1 t ðtÞ tF ðtÞ FðtÞ Σ ≡ ˜ 2 ˜ 2 ˜ 2 þ m ˜ 0 þ 6αλ : ðA5Þ ðmS þ mT þ m 0 Þ; ðA8Þ B0 π 0 1 6α 3 H þ λ0 FðtÞ In the basis the solutions for the soft scalar masses can be captured by relatively simple expressions Δ ≡ ˜ 2 − ˜ 2 HT mH0 mT; ðA6Þ α0 Δ Δ ðtÞ HTðtÞ¼ HT0 0 ; ðA9Þ α0 α0 2 Δ Δ − 2 2 1 − ðtÞ STHðtÞ¼ STH0 mB˜ 0 02 ; ðA10Þ 0 α0 1 2 α0 2 Σ Σ 2 1 − ðtÞ − 2 ¼ STH0 þ m ˜ 0 2 IðtÞ; 1 6α 3 B0 α0 þ λ0 FðtÞ 0 ðA11Þ where we have defined IðtÞ as 2 2 ˜ IðtÞ¼m ˜ 0 IGGðtÞþmB0 Aλ0 IAGðtÞþAλ IAAðtÞ; ðA12Þ B0 0 0 where αλ FðtÞ I t 0 ; AAð Þ¼ 1 6α 2 ðA13Þ ð þ λ0 FðtÞÞ 0 αλ ðtF ðtÞ − FðtÞÞ I t −2 0 ; AGð Þ¼ 1 6α 2 ðA14Þ ð þ λ0 FðtÞÞ FIG. 9. Contours of the IR kinematic suppression of ψ α2 0 − 2 α λ0 ðtF ðtÞ FðtÞÞ λ0 GðtÞ 0 −6 annihilation to mZ as a function of UV couplings. This IGGðtÞ¼ 2 þ ; ðA15Þ 1=2 1 6α 1 6αλ F t suppression is exactly r , for r defined in Eq. (28). ð þ λ0 FðtÞÞ þ 0 ð Þ 075019-16 SIMPLE HIDDEN SECTOR DARK MATTER PHYS. REV. D 102, 075019 (2020) FIG. 10. Each plot depicts the IR coefficient weighting respective soft breaking terms whose linear combination gives the IR value of Σ, defined in Eq. (A17). The axes show the UV couplings that determine the coefficients (up to ϵ-suppressed effects). Σ 2 At one loop, our proxy for the hidden sector scale, , GðtÞ¼− α0ðtÞt2F0ðtÞ; ðA16Þ when evaluated in the IR, can be written in terms of UV π parameters as: 2 2 Σ Σ ˜ ¼ cΣ 0 þ cm ˜ 0 m ˜ 0 þ cm ˜ 0 Aλ mB0 Aλ0 þ cAλ Aλ : ðA17Þ with FðtÞ defined in Eq. (A3). The above equations can be B B0 B 0 0 inverted to yield solutions for the soft masses. We have verified these analytic results against numerical In Fig. 10, we display these coefficients as functions of the solutions of the full two-loop RGE. The two-loop RGEs dimensionless parameters α0 and α in the UV. We note that c and cΣ are the largest numerically, thus we expect UV have previously been discussed in [32]; we correct a few mB˜ 0 small typographical errors: in Equation (A1) of Ref. [32],in specification of the gaugino mass and/or Σ will largely the two-loop part of the β function for the gaugino mass determine the scale of the hidden sector in the IR, absent very α2 α there is an h that should read h; in the two-loop part of the large A-terms in the UV. Furthermore, examination of the 2 β α α values of the c and cΣ in the figure, when taken in concert function for mS there is an S that should read S, and in mB˜ 0 β the two loop expression for the function for m there is an with analogous expressions for the MSSM, see Eq. (24), α that should read αh. shows the disparity between the importance of the UV 075019-17 BARNES, JOHNSON, PIERCE, and SHAKYA PHYS. REV. 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